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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $k \\colon \\R^n \\times \\R^n \\to (-\\infty, +\\infty]$ be a lower semicontinuous function, called a *kernel*.\nLet $\\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\\R^n$ with finite total mass and compact [[support|FQTXV6BQZG]].\nThe *potential of $\\mu$ with respect to $k$* is the function $U_k^\\mu \\colon \\R^n \\to (-\\infty, +\\infty]$,\n$$U_k^\\mu(x) = \\int k(x, y) \\, d\\mu(y).$$\n\n## Remarks\n\nThe integral is well defined: a lower semicontinuous function is bounded below on the compact set $\\set{x} \\times \\operatorname{supp} \\mu$, and $\\mu$ is finite.\nExample: for the Dirac measure $\\mu = \\delta_a$ one gets $U_k^\\mu(x) = k(x, a)$.\nExample: with the [[Newtonian kernel|6TJTWQ5HCR]] on $\\R^3$ and $\\mu$ the uniform probability measure on the unit sphere, $U_k^\\mu(x) = 1$ for $\\abs{x} \\le 1$ and $1/\\abs{x}$ for $\\abs{x} \\ge 1$.\nNon-example: $k(x, y) = -1/\\abs{x - y}$ on $\\R^3$ off the diagonal is not a kernel in this sense.\nIt tends to $-\\infty$ at the diagonal, so no value in $(-\\infty, +\\infty]$ on the diagonal makes it lower semicontinuous.\nThe potential $U_k^\\mu$ is itself lower semicontinuous, by Fatou's lemma.\nStandard kernels are the Newtonian kernel, the [[logarithmic kernel|VDZ5FKYC2T]] and the Riesz kernels $\\abs{x - y}^{\\alpha - n}$ with $0 \u003c \\alpha \u003c n$.\nSigned measures are handled by writing $\\mu = \\mu^+ - \\mu^-$ where both potentials are finite.\n",
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